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The Cut-Elimination Theorem for Differential Nets with Promotion
[chapter]

2009
*
Lecture Notes in Computer Science
*

We extend this analysis to exponential boxes and prove

doi:10.1007/978-3-642-02273-9_17
fatcat:5tep524tprgohnxvsh6jfktn2y
*the**Cut*-*Elimination**Theorem**for**the*whole DiLL: every*differential**net*that is sequentializable can be reduced to a*cut*-free*net*. ...*The*authors have examined*the**cut*-*elimination*of*the**promotion*-free fragment of DiLL by means of a proofnet-like calculus:*differential*interaction*nets*. ...*The**Cut*-*Elimination**Theorem*We prove our main result: switching acyclic*differential**nets*enjoy*the*cutelimination (*Theorem*1). ...##
###
Visible acyclic differential nets, Part I: Semantics

2012
*
Annals of Pure and Applied Logic
*

Visible acyclicity discloses a new kind of correctness

doi:10.1016/j.apal.2011.09.001
fatcat:gkqlo42mcvcopmpfiflbbyvjli
*for**the**promotion*rule of linear logic, which goes beyond sequent calculus correctness. ... (Michele Pagani) 1 Precisely wires also represent*the*identity rules of axiom and*cut*, following*the*spirit of Lafont's interaction*nets*[2] . ... Acknowledgements I am especially grateful to Thomas Ehrhard*for*his several hints which let me understand finiteness spaces and*differential*linear logic. ...##
###
The conservation theorem for differential nets

2015
*
Mathematical Structures in Computer Science
*

We prove

doi:10.1017/s0960129515000456
fatcat:o44phpxvczflboi3jlcra6xg6u
*the*conservation*theorem**for**differential**nets*–*the*graph-theoretical syntax of*the**differential*extension of Linear Logic (Ehrhard and Regnier's DiLL). ...*The*conservation*theorem*states that*the*property of having infinite reductions (here infinite chains of*cut**elimination*steps) is preserved by non-erasing steps. ... In this paper we prove an analogous*theorem**for**differential**nets*-*the*graph-theoretical syntax of*the**differential*extension of Linear Logic (DiLL [ER06] ). ...##
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An Application of Parallel Cut Elimination in Unit-Free Multiplicative Linear Logic to the Taylor Expansion of Proof Nets

2018
*
Annual Conference for Computer Science Logic
*

In order to simulate one

doi:10.4230/lipics.csl.2018.15
dblp:conf/csl/ChouquetA18
fatcat:6pchgimstjeffcndp2rnsjp7pu
*cut**elimination*step in MELL, it is necessary to reduce an arbitrary number of*cuts*in*the**differential**nets*of its Taylor expansion. ... It turns out our results apply to*differential**nets*, because their*cut**elimination*is essentially multiplicative. ...*The*target of Taylor expansion is then in*promotion*-free*differential**nets*[13] , which we call resource*nets*in*the*following, by analogy*with*resource λ-calculus: these form*the*multilinear fragment ...##
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An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets
[article]

2021
*
arXiv
*
pre-print

In order to simulate one

arXiv:1902.05193v6
fatcat:hz55exurljckzly4nhjmimw5ea
*cut**elimination*step in MELL, it is necessary to reduce an arbitrary number of*cuts*in*the**differential**nets*of its Taylor expansion. ... It turns out our results apply to*differential**nets*, because their*cut**elimination*is essentially multiplicative. ... actually taken on*the*occasion of*the*very first workshop of*the*network at Roma Tre in December 2015, dedicated to New trends in linear logic proof-*nets*; and successive versions were presented and discussed ...##
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Confluence of Pure Differential Nets with Promotion
[chapter]

2009
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-642-04027-6_36
fatcat:yt4hntn7w5esjcd4oequg7ee24
*the*confluence of Ehrhard and Regnier's*differential**nets**with*exponential*promotion*, in a pure setting. ... This result generalizes to linear logic regular proof*nets*, where*the*same notion of equivalence was already studied in*the*literature, but only*with*respect to*the*problem of normalization in a typed ... One of these, central to our work, is confluence of*cut**elimination*, i.e.*the*independence of*the*result of*the**cut**elimination*procedure*with*respect to*the*actual*cuts*one decides to reduce. ...##
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An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets

2021
*
Logical Methods in Computer Science
*

In order to simulate one

doi:10.46298/lmcs-17(4:22)2021
fatcat:7d5vr44auvfmhmb6zonjqfgwd4
*cut**elimination*step in MELL, it is necessary to reduce an arbitrary number of*cuts*in*the**differential**nets*of its Taylor expansion. ... It turns out our results apply to*differential**nets*, because their*cut**elimination*is essentially multiplicative. ... actually taken on*the*occasion of*the*very first workshop of*the*network at Roma Tre in December 2015, dedicated to New trends in linear logic proof-*nets*; and successive versions were presented and discussed ...##
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Preface

2016
*
Mathematical Structures in Computer Science
*

of

doi:10.1017/s096012951600044x
fatcat:rdlnum3ddvetxh3p5arwzlp2gy
*the*proof-*net*/term α is itself a (*differential*) proof-*net*/term and an approximation of α. ... and developed much later, like*differentiation*and Taylor expansion*for*proofs. ... Preface 992 proof of*the*completeness*theorem**for*first-order logic. ...##
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Linear Logic and Strong Normalization

2013
*
International Conference on Rewriting Techniques and Applications
*

Strong normalization

doi:10.4230/lipics.rta.2013.39
dblp:conf/rta/Accattoli13
fatcat:niytow7rsbhuxabwthipfbwlti
*for*linear logic requires elaborated rewriting techniques. In this paper we give a new presentation of MELL proof*nets*, without any commutative*cut*-*elimination*rule. ... Moreover, it is an axiomatic proof, as more generally it holds*for*every set of rewriting rules satisfying three very natural requirements*with*respect to substitution, commutation*with**promotion*, full ... Acknowledgements To Delia Kesner, who taught me*the*art of strong normalization proofs, and*with*whom I had uncountable discussions on commutative rules, normalization, and proof*nets*. ...##
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Interpreting a finitary pi-calculus in differential interaction nets

2010
*
Information and Computation
*

To simplify

doi:10.1016/j.ic.2009.06.005
fatcat:4fwmomneuzgwjdihbbqvqenzmi
*the*presentation, these*nets*use only a restricted form of*the**promotion*rule of linear logic, which is sufficient*for*interpreting a replication-free version of*the*π-calculus, as well as ... This shows that*differential*interaction*nets*are sufficiently expressive*for*representing concurrency and mobility, as formalized by*the*pi-calculus. ... They correspond to*the**cut**elimination*rules of*the**differential*linear logic of Section 2.1. ...##
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Logic and linear algebra: an introduction
[article]

2017
*
arXiv
*
pre-print

We give an introduction to logic tailored

arXiv:1407.2650v3
fatcat:hjiw7vripnecdk5352eznu7nwe
*for*algebraists, explaining how proofs in linear logic can be viewed as algorithms*for*constructing morphisms in symmetric closed monoidal categories*with*additional ...*The*interesting part of this vector space semantics is based on*the*cofree cocommutative coalgebra of Sweedler. ...*The*main*theorem*is that*the*relation is weakly normalising, which means:*Theorem*6.7 (*Cut*-*elimination*). ...##
###
Proof Nets and the Linear Substitution Calculus
[article]

2018
*
arXiv
*
pre-print

Nonetheless, two different terms

arXiv:1808.03395v1
fatcat:g4x23aiusvfcxi5dgttxdqoc7q
*with*sharing can still have*the*same proof*nets*representation---a further result is*the*characterisation of*the*equality induced by proof*nets*over terms*with*sharing. ... Here we show that*the*linear substitution calculus, a simple refinement of*the*λ-calculus*with*sharing, is isomorphic to proof*nets*at*the*operational level. ... To*the*reviewers,*for*useful comments. This work has been partially funded by*the*ANR JCJC grant COCA HOLA (ANR-16-CE40-004-01). ...##
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Proof Nets and the Linear Substitution Calculus
[chapter]

2018
*
Lecture Notes in Computer Science
*

Nonetheless, two different terms

doi:10.1007/978-3-030-02508-3_3
fatcat:37gfbyblnfgkxp2o4c2s7eopry
*with*sharing can still have*the*same proof*nets*representation-a further result is*the*characterisation of*the*equality induced by proof*nets*over terms*with*sharing. ... Here we show that*the*linear substitution calculus, a simple refinement of*the*λ-calculus*with*sharing, is isomorphic to proof*nets*at*the*operational level. ... To*the*reviewers,*for*useful comments. This work has been partially funded by*the*ANR JCJC grant COCA HOLA (ANR-16-CE40-004-01). ...##
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Geometry of resource interaction and Taylor–Ehrhard–Regnier expansion: a minimalist approach

2016
*
Mathematical Structures in Computer Science
*

*The*resource λ-calculus is a variation of

*the*λ-calculus where arguments are superpositions of terms and must be linearly used; hence, it is a model

*for*linear and non-deterministic programming languages ... Lastly, we also provide an expanded counterpart of

*the*execution formula, which computes paths as series of objects of GoRI; thus, exchanging determinism and conciseness

*for*linearity and simplicity. ... Path dynamics

*With*

*the*notions just introduced we now define

*the*property of path persistence, that intuitively means "surviving

*cut*-

*elimination*". ...

##
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Proofs as Executions
[chapter]

2012
*
Lecture Notes in Computer Science
*

A type system based on linear logic is used, in which a given process has many different types, each typing corresponding to a particular way of interacting

doi:10.1007/978-3-642-33475-7_20
fatcat:oqbzzkfswfawxagsxkvkqscage
*with*its environment and*cut**elimination*corresponds ... to executing*the*process in a given interaction scenario. ... Section 4 presents*the*proof*nets**for**the*logic of schedulings and its*cut*-*elimination*. ...
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